Solve 78/x+8*7=x

$$7 \frac{ 8 }{ x } +8 \cdot 7 = x$$

Solve for x

$$x=2\sqrt{210}+28\approx 56.982753492$$

$$x=28-2\sqrt{210}\approx -0.982753492$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$7\times 8+8\times 7x=xx$$

Multiply x and x to get x^{2}.

$$7\times 8+8\times 7x=x^{2}$$

Multiply 7 and 8 to get 56. Multiply 8 and 7 to get 56.

$$56+56x=x^{2}$$

Subtract x^{2} from both sides.

$$56+56x-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+56x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 56 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\left(-1\right)\times 56}}{2\left(-1\right)}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\left(-1\right)\times 56}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-56±\sqrt{3136+4\times 56}}{2\left(-1\right)}$$

Multiply 4 times 56.

$$x=\frac{-56±\sqrt{3136+224}}{2\left(-1\right)}$$

Add 3136 to 224.

$$x=\frac{-56±\sqrt{3360}}{2\left(-1\right)}$$

Take the square root of 3360.

$$x=\frac{-56±4\sqrt{210}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-56±4\sqrt{210}}{-2}$$

Now solve the equation x=\frac{-56±4\sqrt{210}}{-2} when ± is plus. Add -56 to 4\sqrt{210}.

$$x=\frac{4\sqrt{210}-56}{-2}$$

Divide -56+4\sqrt{210} by -2.

$$x=28-2\sqrt{210}$$

Now solve the equation x=\frac{-56±4\sqrt{210}}{-2} when ± is minus. Subtract 4\sqrt{210} from -56.

$$x=\frac{-4\sqrt{210}-56}{-2}$$

Divide -56-4\sqrt{210} by -2.

$$x=2\sqrt{210}+28$$

The equation is now solved.

$$x=28-2\sqrt{210} x=2\sqrt{210}+28$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$7\times 8+8\times 7x=xx$$

Multiply x and x to get x^{2}.

$$7\times 8+8\times 7x=x^{2}$$

Multiply 7 and 8 to get 56. Multiply 8 and 7 to get 56.

$$56+56x=x^{2}$$

Subtract x^{2} from both sides.

$$56+56x-x^{2}=0$$

Subtract 56 from both sides. Anything subtracted from zero gives its negation.

$$56x-x^{2}=-56$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+56x=-56$$

Divide both sides by -1.

$$\frac{-x^{2}+56x}{-1}=-\frac{56}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{56}{-1}x=-\frac{56}{-1}$$

Divide 56 by -1.

$$x^{2}-56x=-\frac{56}{-1}$$

Divide -56 by -1.

$$x^{2}-56x=56$$

Divide -56, the coefficient of the x term, by 2 to get -28. Then add the square of -28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-56x+\left(-28\right)^{2}=56+\left(-28\right)^{2}$$

Square -28.

$$x^{2}-56x+784=56+784$$

Add 56 to 784.

$$x^{2}-56x+784=840$$

Factor x^{2}-56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-28\right)^{2}=840$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-28\right)^{2}}=\sqrt{840}$$

Simplify.

$$x-28=2\sqrt{210} x-28=-2\sqrt{210}$$

Add 28 to both sides of the equation.

$$x=2\sqrt{210}+28 x=28-2\sqrt{210}$$